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arXiv:0803.1561

Short Time Dynamics of Scalar Products in Hilbert Space

Adélcio C. Oliveira

Published Mar 11, 2008• 0 citations
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Abstract

We use the semiclassical method proposed in \cite{Adelcio2003} to study scalar products such as the overlap, Husimi functions and fidelity decay. Scars of classical periodic orbits arise naturally in this pertubative expansion. We also derive analytically a well known numerical result that fidelity has a quadratic decay for short times. We study the overlap in the chaotic regime and integrable for some simple systems.

DOI
10.48550/arxiv.0803.1561

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We use the semiclassical method proposed in \cite{Adelcio2003} to study scalar products such as the overlap, Husimi functions and fidelity decay. Scars of classical periodic orbits arise naturally in this pertubative expansion. We also derive analytically a well known numerical result that fidelity has a quadratic decay for short times. We study the overlap in the chaotic regime and integrable for some simple systems.
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.
The Husimi function provides a phase-space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate-density momentum-density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.
Just as a coherent state may be considered as a quantum point, its restriction to a factor space of the full Hilbert space can be interpreted as a quantum plane. The overlap of such a factor coherent state with a full pure state is akin to a quantum section. It defines a reduced pure state in the cofactor Hilbert space. Physically, this factorization corresponds to the description of interacting components of a quantum system with many degrees of freedom and the sections could be generated by conceivable partial measurements. The collection of all the Wigner functions corresponding to a full set of parallel quantum sections defines the Husimi–Wigner representation. It occupies an intermediate ground between the drastic suppression of non-classical features, characteristic of Husimi functions, and the daunting complexity of higher dimensional Wigner functions. After analysing these features for simpler states, we exploit this new representation as a probe of numerically computed eigenstates of a chaotic Hamiltonian. Though less regular, the individual two-dimensional Wigner functions resemble those of semiclassically quantized states.
The quantisation of the two-dimensional toric and spherical phase spaces is considered in analytic coherent state representations. Every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its Husimi function. For eigenstates of quantised systems, this description explicitly reflects the nature of the underlying classical dynamics: in the semiclassical regime, the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems. This multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems.